It happens the entire game. The second number you set on Live Chess is how many "Bonus" seconds the player gets on their turn. The "Bonus" seconds were designed to give chess players time to write down their notations.
Time increments
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It might also be simply lag. For example, your opponent might have 30 seconds left and take 5 seconds to make a move. Your computer receives the moves 2 seconds late, so your computer showed their clock tick down to 23 seconds. When the move is received, your computer corrects it back up 2 seconds and thus it appears as an increment although your opponent actually just didn't use that time.
Why does the site add increments when someone gets into their last minute of time? Is this happening to anyone else? I find it extremely annoying.